| L(s) = 1 | − 9·3-s + 25·5-s + 81·9-s + 238·11-s − 142·13-s − 225·15-s + 98·17-s + 862·23-s + 625·25-s − 729·27-s − 238·29-s − 1.44e3·31-s − 2.14e3·33-s − 1.58e3·37-s + 1.27e3·39-s − 1.77e3·43-s + 2.02e3·45-s + 3.26e3·47-s + 2.40e3·49-s − 882·51-s + 5.95e3·55-s − 2.64e3·59-s − 3.55e3·65-s + 8.30e3·67-s − 7.75e3·69-s − 5.62e3·75-s + 1.10e4·79-s + ⋯ |
| L(s) = 1 | − 3-s + 5-s + 9-s + 1.96·11-s − 0.840·13-s − 15-s + 0.339·17-s + 1.62·23-s + 25-s − 27-s − 0.282·29-s − 1.50·31-s − 1.96·33-s − 1.15·37-s + 0.840·39-s − 0.961·43-s + 45-s + 1.47·47-s + 49-s − 0.339·51-s + 1.96·55-s − 0.758·59-s − 0.840·65-s + 1.84·67-s − 1.62·69-s − 75-s + 1.76·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.066042333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.066042333\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{2} T \) |
| 5 | \( 1 - p^{2} T \) |
| good | 7 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 11 | \( 1 - 238 T + p^{4} T^{2} \) |
| 13 | \( 1 + 142 T + p^{4} T^{2} \) |
| 17 | \( 1 - 98 T + p^{4} T^{2} \) |
| 19 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 23 | \( 1 - 862 T + p^{4} T^{2} \) |
| 29 | \( 1 + 238 T + p^{4} T^{2} \) |
| 31 | \( 1 + 1442 T + p^{4} T^{2} \) |
| 37 | \( 1 + 1582 T + p^{4} T^{2} \) |
| 41 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 43 | \( 1 + 1778 T + p^{4} T^{2} \) |
| 47 | \( 1 - 3262 T + p^{4} T^{2} \) |
| 53 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 59 | \( 1 + 2642 T + p^{4} T^{2} \) |
| 61 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 67 | \( 1 - 8302 T + p^{4} T^{2} \) |
| 71 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 73 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 79 | \( 1 - 11038 T + p^{4} T^{2} \) |
| 83 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 89 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 97 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.44182909611730297786342882265, −9.458741579983557474911626899583, −9.004400343534309654910954997230, −7.18639736322355508150865033688, −6.69439250811195691643205019063, −5.67309173812592399014674876978, −4.89491425229972795839588312229, −3.63720725079845003573292503390, −1.89440435103203811613123405465, −0.897714562490923954675386431528,
0.897714562490923954675386431528, 1.89440435103203811613123405465, 3.63720725079845003573292503390, 4.89491425229972795839588312229, 5.67309173812592399014674876978, 6.69439250811195691643205019063, 7.18639736322355508150865033688, 9.004400343534309654910954997230, 9.458741579983557474911626899583, 10.44182909611730297786342882265
